求y=ax的平方-2ax+3(a不等于0) 在x大于等于a 小于等于2a+3 内的最值
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求y=ax的平方-2ax+3(a不等于0) 在x大于等于a 小于等于2a+3 内的最值
求y=ax的平方-2ax+3(a不等于0) 在x大于等于a 小于等于2a+3 内的最值
求y=ax的平方-2ax+3(a不等于0) 在x大于等于a 小于等于2a+3 内的最值
y=ax^2-2ax+3
=a(x-1)^2-a+3,
a≤x≤2a+3
a-1≤x-1≤2a+2
1.
2a+2<0时,即a<-1
a-1≤x-1≤2a+2
4(a+1)^2≤(x-1)^2≤(a-1)^2
a(a-1)^2≤a(x-1)^2≤4a(a+1)^2
a(a-1)^2-a+3≤a(x-1)^2-a+3≤4a(a+1)^2-a+3
a(a-1)^2-a+3≤y≤4a(a+1)^2-a+3;
2.
a-1>0时,即a>1
a-1≤x-1≤2a+2
(a-1)^2≤(x-1)^2≤4(a+1)^2
4a(a+1)^2≤a(x-1)^2≤a(a-1)^2
4a(a+1)^2-a+3≤a(x-1)^2-a+3≤a(a-1)^2-a+3
4a(a+1)^2-a+3≤y≤a(a-1)^2-a+3;
3.
a-1<0,2a+2>0时,即-1<a<0或0<a<1
(1)-1<a<0且2a+2>-(a-1),即-1/3<a<0
a-1≤x-1≤2a+2
0≤(x-1)^2≤4(a+1)^2
4a(a+1)^2≤a(x-1)^2≤0
4a(a+1)^2-a+3≤a(x-1)^2-a+3≤-a+3
4a(a+1)^2-a+3≤y≤-a+3;
(2)-1<a<0且2a+2<-(a-1),即-1<a<-1/3
a-1≤x-1≤2a+2
0≤(x-1)^2≤(a-1)^2
a(a-1)^2≤a(x-1)^2≤0
a(a-1)^2-a+3≤a(x-1)^2-a+3≤-a+3
a(a-1)^2-a+3≤y≤-a+3;
(3)0<a<1且2a+2>-(a-1),即0<a<1
a-1≤x-1≤2a+2
0≤(x-1)^2≤(a-1)^2
0≤a(x-1)^2≤a(a-1)^2
-a+3≤a(x-1)^2-a+3≤a(a-1)^2-a+3
-a+3≤y≤a(a-1)^2-a+3;
(4)0<a<1且2a+2<-(a-1),无解.
综上所述
a<-1时,a(a-1)^2-a+3≤y≤4a(a+1)^2-a+3;
-1<a<-1/3时,a(a-1)^2-a+3≤y≤-a+3;
-1/3<a<0时,4a(a+1)^2-a+3≤y≤-a+3;
0<a<1时,-a+3≤y≤a(a-1)^2-a+3;
a>1时,4a(a+1)^2-a+3≤y≤a(a-1)^2-a+3.